Gas permeation through graphdiyne-based nanoporous membranes

Nanoporous membranes based on two dimensional materials are predicted to provide highly selective gas transport in combination with extreme permeance. Here we investigate membranes made from multilayer graphdiyne, a graphene-like crystal with a larger unit cell. Despite being nearly a hundred of nanometers thick, the membranes allow fast, Knudsen-type permeation of light gases such as helium and hydrogen whereas heavy noble gases like xenon exhibit strongly suppressed flows. Using isotope and cryogenic temperature measurements, the seemingly conflicting characteristics are explained by a high density of straight-through holes (direct porosity of ∼0.1%), in which heavy atoms are adsorbed on the walls, partially blocking Knudsen flows. Our work offers important insights into intricate transport mechanisms playing a role at nanoscale.

To calibrate gas permeability measured by the helium leak detector, we note that it allows internal calibration for the flow rate of 4 He. As a control experiment to demonstrate the accuracy of calibration, we tested the permeation of 4 He through a 1-μm (in diameter) aperture in silicon nitride. The leak detector correctly measured permeability Γ * 4He that approaches (error < 10%) its theoretical value of , where L is the length of the aperture. In these measurements, the pressure applied is kept < 0.1 bar to ensure the Knudsen condition (mean free path λ > aperture dimension d0, as discussed in the main texts) is fulfilled. For the calibration of other gas species, a correction factor was calculated from the ratio between their measured permeability and that of 4 He (more specifically, 4 * × √ 4 0 , with m4He and m0 the atomic mass of 4 He and gases under investigation, respectively) through the aperture. Similar procedures were applied to the calibration of gas permeation measured by the mass spectrometer, using the same 1-μm aperture. For cryogenic temperature measurements. Indium seals (black empty circles next to the SiN wafer) were used instead of rubber O-rings which might fail at low temperatures.

Gas permeation through bare-hole devices
The permeance of bare-hole devices shown in Fig. 2b were measured at low feed pressure, where the mean free path of gas molecules is sufficiently large such that Knudsen flow occurs even for micro-meter sized aperture (Supplementary Figure 5a). That ensured the accuracy of our porosity estimation. In the same Knudsen regime for graphdiyne and 2 μm hole, gas flow conductance Ca through thin aperture depends linearly on the rate of impingement of molecules over pore area A ( = √ 2 ) 6 . The permeance through bare hole is ~1000 times higher than that of the graphdiyne membrane, can be therefore translated using the standard definition as 0.1% porosity.
Helium permeability through a thin aperture of 50 nm diameter in size were also measured.

DFT simulations
For the density functional theory-based first-principles calculations, the projector augmented wave 7 method was used in order to portray the pseudopotentials of C atom and the noble gas elements as implemented in Vienna Ab-initio Simulation Package (VASP) 8 . The exchangecorrelation potential was taken into account by considering the local density approximation (LDA) within the Perdew-Burke-Ernzerhof (PBE) form 9 . For the geometry optimizations a kinetic energy cutoff of 500 eV was used for the plane-wave basis. The convergence criterion of the total force on each atom was reduced to 10 -5 eV/Å and the convergence criterion of the energy was set to 10 -6 eV.
The primitive unit cell of graphdiyne structure is composed of 18 C atoms and it was optimized using a k-point mesh of 18 × 18 × 1. In order to simulate the propagation of noble gas atoms through the hole of the graphdiyne structure, a super cell containing 72 C atoms was used. We checked that this was sufficiently large to avoid inter simulation cell interaction. The initial distance of a noble gas atom to the basal plane of graphdiyne was taken to be 8 Å and the vertical position of the noble gas atom was changed by 0.5 Å steps and then the total energy was calculated at each step. Note that while the rigid structure refers to the case when all the carbon atoms are frozen, the relaxed case refers to fully optimization of the whole structure as gas atoms diffuse through the lattice rings in graphdiyne. He-Xe separation. Solid black lines indicate the state-of-art boundary, determined by the authors using gas permeation data from literatures. Filled red star represents the separation performance from the data of single component gas transport through graphdiynebased membranes; for binary mixtures, both the permeance and selectivity are corrected by about 50%, as shown by the empty red star in (b). The literature data include membranes made from polymers [10][11][12][13] , zeolites [14][15][16][17][18][19][20] , carbon nanotubes 21 , porous organic cages 22 , metal organic frameworks (MOF) 23,24 , silicate 25 , carbon molecular sieves 26 and porous alumina 17 . All data points are experimental results with no corrections/assumptions made to membrane thickness and pore density. The figure also includes results from previously reported porous graphene. For those of perforated graphene with single-nanopores 27 , we assume their pore density of 10 12 cm -2 (which is the highest density currently achievable by top-down methods) without loss of single-pore functionality.